Associate Professor Timothy Trudgian, UNSW Canberra
Valeriia Starichkova, UNSW Canberra
Dr Michaela Cully-Hugill, UNSW Canberra
We will cover a several topics in Number Theory, the proof of the Dirichlet theorem and discussions around primes in arithmetic progressions. Participants will develop their problem-solving skills through various exercises and motivating discussions during the workshops. Particpiants will get acquainted with some of the basic techniques used nowadays in analytic number theory.
- Elementary proofs about primes in arithmetic progressions
- Introduction to primes, primes congruent to 1 modulo and integer k
- Elementary attempts to prove the Dirichlet theorem via methods by Chebyshev and Erd˝os.
- Obstructions of elementary methods
- Introduction to complex analysis and analytic techniques covering the Riemann zeta-function and its extension to the complex plane using the functional equation
- An analytic proof of the infinitude of prime numbers.
- Introduction to the proof of the Dirichlet theorem: the Dirichlet characters and the properties of the Dirichlet L-functions.
- Finalising the proof of the Dirichlet theorem
- More questions around primes represented by polynomials and primes in arithmetic progressions like the Green-Tao theorem
- Arithmetic progressions; acquaintance with modular arithmetic; complex numbers; one-variable calculus
- A basic understanding of complex analysis would be helpful but not required.
Participation in all lectures and tutorials is expected.
For those completing the subject for their own knowledge/interest, evidence of at least 80% attendance at lectures and tutorials is required to receive a certificate of attendance.
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